At the early stage of an emerging infectious disease outbreak, the interplay between individual protective behavior and therapeutic resources profoundly influences the effectiveness of epidemic control. By extending the classic behavior–disease coupling framework, this paper developed a dynamic model that concurrently incorporates heterogeneous behavioral adoption dynamics and saturation effects in treatment resources. Two behavioral regulation parameters were introduced to quantify their synergistic influence on transmission dynamics. Theoretical analysis yielded the basic reproduction number for both homogeneous and mixed systems, thereby establishing the threshold dynamics of the model. Numerical simulations demonstrated that increasing the intensity of behavioral adoption significantly reduces both the infection scale and transmission risk, and can even lower R₀ below the epidemic. However, if the driving force for adoption is insufficient, transmission risk in the mixed system may exceed that in a system with no behavioral adoption, suggesting that ineffective interventions could exacerbate disease spread. Under resource constraints, our results indicate that large-scale, high-compliance behavioral interventions are essential for epidemic control. They further provides a mathematical basis for designing integrated strategies that balance “behavioral maintenance” and “resource allocation”.

This paper deals with a second-order fully differential equation involving a ϕ−Laplacian operator, which generalizes the more common p-Laplacian, with a parameter, applied to a nonlinearity depending on the unknown function and its first derivative. The main result states the problem’s solvability for parameter values for which there are lower and upper solutions for the problem. Singular ϕ−Laplacians considered with an application to the special relativity field.

This study addresses an inverse source problem for a general stationary kinetic equation involving absorption and scattering terms. The primary objective is the simultaneous reconstruction of the phase space particle distribution and an unknown source in the model, utilizing boundary conditions defined by derivatives and additional interior measurements. In the theoretical part of the study, the uniqueness of the solution to the inverse problem is proved. For the numerical solution, a comprehensive computational framework is developed based on finite difference approximations for derivatives and the trapezoidal rule for the discretization of the integral terms. The reconstruction strategy employs bilinear interpolation techniques coupled with Tikhonov regularization. In addition, a Monte Carlo noise sensitivity analysis is carried out to assess the stability and robustness of the proposed method under perturbed data. Finally, the effectiveness of the method is illustrated through numerical examples.

Discrete-time modeling and reliable control of mechanical systems with inertia, spatial coupling, time delays, and random disturbances are of fundamental importance in mechanism and machine science, particularly for safety-critical and robotic applications. This paper investigated the nonlinear dynamics and stabilization of a class of discrete-time inertial reaction–diffusion mechanical systems subject to stochastic perturbations and time-varying delays. A strong spatio-temporal discretization framework was employed to formulate the system as second-order difference equations incorporating inertial effects, diffusive coupling, delayed interactions, and stochastic excitations. To describe long-term dynamic behavior in noisy environments, a mean-square pseudo almost periodic solution concept was introduced. Sufficient conditions were established to ensure the existence and uniqueness of such solutions via fixed-point techniques. Moreover, a feedback control strategy was developed to achieve global exponential stabilization in the mean-square sense. Explicit stability criteria were derived, revealing the effects of inertia, diffusion intensity, delay bounds, stochastic disturbance levels, and control gains on system reliability and convergence performance. Numerical simulations on a multi-joint robotic manipulator validated the theoretical results and illustrated the trade-off between convergence speed and robustness under stochastic disturbances.

This work studies the nonlinear dynamics of plates vertically attached to a channel using the immersed boundary-lattice Boltzmann-general interpolation material point method. The hydrodynamic flow and dynamics of the plate walls over a broad range of Reynolds numbers, structure-to-fluid mass ratio, and bending stiffness are analysed. Four modes are identified: stable plate with symmetric vortex, stable plate with asymmetric vortex, unstable plate with symmetric vortex, and unstable plate with asymmetric vortex. The modes represent distinct flow-structure interaction scenarios observed during the study of plates subjected to fluid flows. A symmetric wake remains aligned with the centreline of the artery over time, while an asymmetric wake deviates upwards or downwards from the centreline. The potential pathways to steady and unsteady behaviour are explored. The physical mechanisms that cause different modes of dynamic response for the plates are discussed.

The main goal of this paper was to develop new skills to investigate the blow-up properties of solutions to an initial-boundary value problem for a p-Laplacian-type pseudo-parabolic equation with singular potential and logarithmic source. By establishing a crucial reverse Sobolev inequality, we derived that the solutions blow up in finite time under a new blow-up criterion and we estimated the lifespan of the weak solutions from both above and below. It is worthy to point out that our blow-up criterion implies that this problem admits finite time blow-up solutions at an arbitrarily high initial energy level. From methods to results, we partially extended some results obtained in recent literatures.

This paper is concerned with the quasilinear Kirchhoff-Schrödinger-Poisson system with critical growth. By using the Krasnoselskii genus theory and variational methods, we prove the existence of infinitely many solutions for this system.

Aspect-based sentiment analysis (ABSA) aims to identify and classify sentiment polarities toward specific aspects in text. To address the limitations of existing models, such as oversimplified structures and underutilized fine-grained features, we propose an ABSA model that fuses multi-granularity features with hierarchical networks. Through the construction of a multi-granularity feature extraction module, we employed a graph convolutional network (GCN) on constituent and dependency trees to capture syntactic and semantic information, augmented with external knowledge. This enables comprehensive feature extraction from four granularity levels: constituent structure, dependency relations, contextual cues, and external knowledge. To effectively fuse these diverse features, we designed a multi-layer feature fusion network (MLFF). Utilizing cross attention and orthogonal projection, the MLFF module iteratively refines feature interactions. Extensive experiments on the SemEval 2014 and Twitter datasets show that our model outperforms existing ABSA models and can effectively enhance task performance.

Fishing moratoriums are vital for fishery management, restoring fish stocks, safeguarding ecosystems, and ensuring sustainable development. In this study, we explored and studied a canonical model with a Holling type-III functional response consisting of two sub-equations switching each other. If the length of the closed season $\bar{T} \leq T^{*}$, the trivial steady state is globally asymptotically stable. For $\bar{T}>T^{*}$, we provided a complete analysis of the existence of a unique periodic solution and three exact periodic solutions. We also provided sufficient conditions for the existence of a globally asymptotically stable periodic solution. Finally, numerical examples were provided to illustrate the theoretical results.

Learning conditional dependence structure among latent concepts from unstructured text can provide an interpretable graph prior for downstream models. Existing methods estimate such a concept graph by modeling GloVe-style concept embeddings as matrix-normal observations with sparse row/column precision matrices, but its Gaussian assumption is usually not satisfied for modern neural embeddings. In this paper, we extend this framework to a nonparanormal matrix-normal graphical model, allowing monotone marginal transformations while preserving the conditional- independence interpretation of the row/column precision matrix. We construct matrix-valued concept embeddings from language model representations and estimate sparse row/column precision matrices via a penalized likelihood that replaces sample covariances with rank-based correlation estimates, improving the robustness in cases with skewness and heavy tails. The resulting concept graph is defined by the support of the row precision matrix and is used as a structural prior for downstream prediction. Experiments on a salary-prediction task show that the proposed nonparanormal precision graph consistently reduces absolute-error risk and improves stability over competitors. Ablations further indicate that performance gains stem from meaningful conditional-dependence structure, rather than merely from increased model capacity.

To effectively capture the nonlinear and complex patterns inherent in small-sample data, this paper proposed a novel self-adaptive second-order discrete grey model incorporating new information priority accumulation. The modeling mechanism and theoretical properties of the proposed model were systematically examined. To mitigate multicollinearity and enhance the stability of parameter estimation in small-sample contexts, Ridge regression regularization was strategically incorporated into the modeling framework. The computational efficiency of the parameter optimization was demonstrated through comparative benchmarking, which substantiated the superiority of the differential evolution algorithm over conventional methods. Furthermore, the model’s robustness and stability were verified via Monte Carlo simulations and sensitivity analyses with varying training set proportions. The proposed model was applied to forecast China’s total electricity generation as well as generation from specific energy sources. The results demonstrated that the proposed model achieves higher forecasting accuracy than benchmark models. Finally, the power generation across four modes from 2025 to 2030 was predicted and analyzed.

A novel approach for the high-precision evaluation of hypersingular integrals on a circle by the spline approximation of the periodic density function is presented. A cubic spline interpolation function with periodic boundary conditions, enforced through a cyclic tridiagonal system, is constructed through the uniform partitioning of the periodic interval. Through the analytical properties of the Clausen functions, an explicit expression for the integral is derived, and a rigorous error analysis is conducted. Theoretical results demonstrate that a convergence rate of O(h³) at non-superconvergent points and O(h⁴) superconvergence at the zeros of the special function Φ(τ) are attained. It is further demonstrated that the superconvergence phenomenon is uniformly discerned whenever the singular point coincides with the zeros of Φ(τ), regardless of the singular point’s relative position within the mesh. Finally, a numerical example is presented for illustrating the effectiveness of the proposed method. The computed errors across diverse mesh sizes and singular point locations are in remarkable agreement with theoretical predictions.

In this paper, we develop a spatial heterogeneous compartment model to investigate the combined impacts of nonlocal diffusion and antiretroviral therapy (ART) on the transmission dynamics of human immunodeficiency virus/acquired immunodeficiency syndrome (HIV/AIDS) within men who have sex with men (MSM). We have overcome the difficulty caused by the lack of compactness of the nonlocal operator. With the updated equation, we obtain the function formulation of the next generation operator R. Then, the basic reproduction number R₀ is obtained, which is the spectral radius of R. Concerning analytical investigation, according to Lipschitz continuity assumption of the parameters, the linearized model under the assumption of no disease at steady state has a dominant eigenvalue associated with a non-negative eigenfunction with positive values. By specifying the eigenfunction to be the integral kernel of the Lyapunov function, we demonstrate that, for R₀ < 1, as time progresses, all solutions approach the disease-free steady state globally. Based on the uniform persistence theory applicable to dissipative systems, it is proved that when R₀ > 1, uniform persistence is maintained in the model, which implies the presence of at least one positive equilibrium. In the numerical simulation part, the theoretical results are verified, and these findings indicate that the MSM population experiences significantly reduced HIV transmission rates through effective ART implementation.

The rapid expansion of expressway open-toll plazas, where high-speed electronic toll collection lanes operate alongside slower manual and single-lane toll lanes, has introduced complex crash risks that are difficult to capture using traditional crash-frequency models alone. Sparse crash records, heterogeneous operating conditions, and evolving tolling configurations call for data-driven approaches that integrate simulation, surrogate safety metrics, and computational learning. This study aims to perform a comprehensive risk assessment of open-toll plaza configurations by combining surrogate safety measure (SSM) estimations with computational learning (CL)-based accident prediction models. First, a broad set of geometric and operational scenarios is generated by systematically varying lane layout, toll-lane composition, merging length, speed limit, traffic volume, and heavy-vehicle share. For each scenario, microscopic traffic simulations are conducted, and SSMs-including time to collision (TTC), post-encroachment time (PET), and conflict counts-are extracted to quantify instantaneous interaction risk at the vehicle level. These SSMs serve both as comparative safety indicators and as target outputs for subsequent CL models. Next, the CL model is trained to predict conflict frequencies from design and traffic features, while a complementary parametric count model is estimated for benchmarking. To enhance interpretability, eXplainable AI (XAI) attribution techniques are used to decompose the CL predictions into feature-level contributions, revealing nonlinear and interaction effects associated with high-risk operating regimes. The results highlight the dominant influence of traffic volume and toll lane ratio on conflict occurrence. By integrating SSM-based simulation outputs with CL and XAI, the proposed approach provides quantitative and interpretable evidence that supports safer design and operation of expressway open-toll plazas within next-generation data-driven transportation systems.

In this paper, we systematically characterize the types of oscillator algebras, ranging from the simplest case $\mathfrak{D}_{4}$ to the most general $\mathfrak{D}_{2 m+2}^{\lambda}$. Subsequently, we construct multiple faithful matrix representations and vector field representations for $\mathfrak{D}_{2 m+2}^{\lambda}(\mathbb{R})$, and give a non-go theorem for a certain representation of $\mathfrak{D}_{2 m+2}^{\lambda}(\mathbb{C})$. Furthermore, leveraging the faithful representation of $\mathfrak{D}_{2 m+2}^{\lambda}(\mathbb{R})$ via the special symplectic Lie algebra $\mathfrak{s p}_{2 m+2}(\mathbb{R})$, we establish a Leibniz algebra associated with $\mathfrak{D}_{2 m+2}^{\lambda}(\mathbb{R})$.

Fuzzy relation matrices are a core representation mechanism in fuzzy inference and fuzzy system modeling, yet practical data are often raw multichannel sensor readings rather than pre- defined input/output fuzzy vectors. We develop an end-to-end supervised learning framework that (i) maps sensor readings to fuzzy vectors via per-channel Gaussian fuzzification and (ii) learns a max- min fuzzy relation matrix from adjacent-time sensor pairs. To overcome the nonsmooth max-min composition, we introduce a differentiable softmax/softmin surrogate with temperature annealing and derive a stochastic-gradient training procedure that jointly optimizes the relation matrix and fuzzifier parameters. Experiments on structured synthetic sensor streams show stable convergence and improved structure recovery over a fixed (data-driven) fuzzifier baseline while approaching an oracle fuzzifier.

A fractional-order model (FOM) was developed to investigate plant disease transmission (PDT) through a system of dimensionally consistent fractional differential equations (FDEs) with the Caputo derivative. The model’s well-posedness was established by proving existence and uniqueness of solutions via a fixed-point theory and the contraction mapping principle. Positivity, boundedness, and the equilibrium points (EPs) of the system were then characterized, followed by an analysis of their local and global stability using the Routh-Hurwitz criteria and LaSalle’s invariance principle. The control reproduction number (R_c) was derived using the next-generation matrix method, and a sensitivity analysis highlighted the parameters most influential to disease spread. A fractional optimal control problem (FOCP) incorporating preventive and curative time-dependent interventions was formulated, and necessary optimality conditions (NOCs) were obtained through a kind of Pontryagin’s maximum principle (PMP). The resulting optimality system was solved numerically using a forward- backward sweep method (FBSM) based on the fractional Euler scheme, enabling the evaluation of control strategies. Three optimal intervention strategies emerged, each shaping the epidemic trajectory differently depending on the distinguishing parameter ε in the two-stage transmission process. Numerical simulations depicted the behavior of R_c across different ε and fractional order α, while tabulated objective functional values exhibited the efficacy of the proposed controls. Overall, the framework offered practical insights for mitigating and potentially eliminating plant epidemics under diverse control strategies.

This paper proposes a novel stochastic two-patch model that incorporates a Beverton–Holt-type nonlinear birth rate and dispersal driven by an Ornstein–Uhlenbeck process. We first establish the existence and uniqueness of a global positive solution, thereby ensuring the biological well-posedness of the model. By constructing appropriate Lyapunov functions and applying Itô’s formula, we further demonstrate the existence of a stationary distribution, which characterizes the long-term statistical behavior of the system. Sufficient conditions for population extinction are then derived, providing criteria for predicting when the population is destined to disappear. We perform numerical simulations using the Euler–Maruyama method to validate the main theoretical results. Finally, as an empirical application, the stochastic model achieves a good fit to zooplankton data, confirming its practical validity and offering important insights for mathematical ecology, as well as for species conservation, population regulation, and resource management in fragmented habitats.

This paper presents a complete classification of conformal Ricci collineations (CRCs) associated with the Bott connection on three-dimensional Lorentzian Lie groups. For each of the seven distinct classes of such groups, we derive explicit conditions under which left-invariant vector fields define CRCs, and we characterize the corresponding solution spaces. Specifically, we find that groups G₁ and G₇ admit rich non-trivial CRCs associated with ∇^B1, highlighting the rich geometric structure induced by this connection.

This paper presents novel safe reinforcement-learning-based, fixed-time control framework for a class of discrete time uncertain nonlinear systems which guarantees fixed-time stability (FxTS). This framework is developed to ensure fixed-time convergence for nonlinear systems in the absence of exact model. The main contribution is to present a unified framework by integration of fixed-time Lyapunov-based constraints and Gaussian process-based uncertainty modeling into the safe reinforcement learning loop in a manner that each candidate policy must ensure the fixed-time Lyapunov decrement condition within a uniform bounded convergence time in the entire learning process. The Lyapunov function for the nominal fixed-time stable closed loop system is used for enlarging the safe set for FxTS with accumulation of system data through refining the traditional Gaussian process model for uncertain system dynamics Gaussian process model is used to learn the unknown system dynamics online and to provide probabilistic confidence bounds. These bounds are directly incorporated into the fixed-time Lyapunov condition, leading to safety guarantees and monotonic expansion of the Lyapunov certified safe set. The control policy is then optimized for the enlarged safe set through exploration while ensuring FxTS. The proposed approach is validated through simulated results of inverted pendulum and reduced-order vehicle dynamic model for lateral vehicle dynamics. The validation for both nonlinear systems confirms fixed-time convergence, certified expansion of safe operating region, and improved control performance under safety and Lyapunov constraints.

The phase-field crystal (PFC) model is widely used to describe atomic-scale crystalline patterns on diffusive time scales and to study the long-time evolution of microstructures. From the viewpoint of computational fluid dynamics, the standard two-mode PFC formulation for face-centered cubic ordering (PFC–FCC model) can be regarded as a high-order phase-field model for solid–liquid phase transitions, in which the motion of diffuse solid–liquid and grain boundaries is encoded in the evolution of a conserved order parameter. In this work, we constructed a linear, fully decoupled time-discretization scheme for the PFC-FCC system based on the explicit Invariant Energy Quadratization (EIEQ) approach. The scheme requires only the solution of constant-coefficient elliptic problems at each time step, making it well suited for fast solvers in large-scale simulations. We proved unique solvability, unconditional energy stability, and a rigorous optimal first-order a priori error estimate under suitable regularity assumptions, relying on a uniform L^∞-bound for the discrete density to control the nonlinear terms. Numerical tests with a manufactured solution confirmed the predicted convergence rate, while two- and three-dimensional simulations of solidification and crystallization with moving crystalline interfaces illustrated the stability, energy-dissipation property, and effectiveness of the EIEQ-based scheme as a numerical tool for long-time simulations of FCC/BCC pattern selection.

In this paper, we propose an HBV infection model with DNA-containing capsids and age structures in both the latent and infected compartments. We derive the basic reproduction number ℜ₀ and prove, via Lyapunov functions and LaSalle’s invariance principle, that the global dynamics are fully determined by ℜ₀: viral clearance when ℜ₀ < 1 and endemic persistence when ℜ₀ > 1. The local stability of each equilibrium is rigorously established by employing linearization techniques and analyzing the associated characteristic equations. Compared to the previous age-structured HBV models without a latent compartment, our results reveal the significant impact of latent age structure on infection thresholds and disease progression.

A structured group is a group endowed with a binary internal relation compatible with the group operation, which can be generalized to a type of partially ordered group. In this paper, we investigated two kinds of structured groups: first, those in which the binary internal relation is an equivalence relation, where we showed that the quotient set acquires a group structure, leading to a characterization of its normal subgroups. As a consequence, we obtained that group quotients by congruencies are equivalent to group quotients by normality. Additionally, we proved that the quotient set of a topological group with a compatible equivalence relation is also a topological group whose canonical projection is an open map. The second type of structured groups, we considered, are those with a partial order relation. In this case, we identified nonrestrictive sufficient conditions for the order topology to define a monoid topology, hence, a group topology.

The frequency-selective response of the nervous system bears a strong resemblance to filtering mechanisms. However, how concurrent cross-modal filtering influences neuronal synchronization remains unclear. This study constructed a coupled photosensitive-auditory neuron model incorporating flux-controlled memristors. Through inductive coil connections, we investigated its synchronous dynamics under frequency-swept signal excitation. The key innovation lies in simultaneously applying swept chirp excitation to both light and sound inputs, achieving biophysically plausible bandpass filtering based on the intrinsic frequency tuning of sensory pathways, while employing an adaptive coupling strategy based on Hamiltonian energy differences. Results demonstrate that bimodal filtering significantly enhances frequency-dependent synchronization. When the sweeping signal traverses the dual passbands, the system exhibits robust synchronization characterized by phase locking and energy minimization, significantly outperforming unfiltered or unimodal conditions. This study reveals cross-modal collaborative filtering as a fundamental mechanism for neural synchronization, providing a computational framework for multisensory integration. It offers significant implications for developing novel neural prostheses synchronized by photoacoustic cues.

This paper focuses on the regularity criteria for the 3D generalized magnetohydrodynamic (MHD) equations, particularly those involving the partial components of the velocity u₃ and the magnetic field (b₁,b₂,b₃). For any l,i,j,k ∈{1, 2, 3}, we establish regularity criteria via ∂_l u₃ and (∂_i b₁, ∂_j b₂, ∂_k b₃), u₃ and (∂_i b₁, ∂_j b₂, ∂_k b₃), or ∂_l u₃ and (b₁,b₂,b₃). Furthermore, a comprehensive regularity condition is obtained via (ξu₃, ηb₁, ζb₂, σb₃) and (ξ^′∂_l u₃, η^′∂_i b₁, ζ^′∂_j b₂, σ^′∂_k b₃) with ξ + ξ^′ = η + η^′ = ζ + ζ^′ = σ + σ^′ = 1 and ξ, η, ζ, σ, ξ^′, η^′, ζ^′, σ^′ ∈{0, 1}.

This paper deals with the stability issue of aperiodic sampled-data control systems with fixed delay. First, to obtain the less conservative stability criterion, the hierarchical looped-functional is assembled. In the functional, the integrals in the integral inequalities are directly utilized in the construction. Hence, the integrals of the system equation are prevented to avoid the products of the free matrices and system matrices. Also, the integrals over the segmented interval [t,t − γ] are introduced. Then, by employing the generalized free-matrix-based integral inequalities, the integrals of the same quadratic product term over different time intervals are estimated to acquire the linear functions of these intervals. Next, the hierarchical stability conditions are presented in terms of the linear matrix inequalities based on the Schur complement. Finally, some numeric examples are used to illustrate the superiority of the proposed method.

Sclerodermus guani is a facultative natural enemy with dual functional responses of predation and parasitism, which has been widely used in the biological control of Monochamus saltuarius, the vector insect of Pine Wilt Disease (PWD). However, most existing studies treat its predation and parasitism behaviors in isolation and ignore the synergistic effect of the dual functions. Based on this, we innovatively construct a facultative predation-parasitism reaction-diffusion model with top-hat nonlocal perception, dual predation-parasitism functional responses, and parasitic time delay, and we systematically analyze the spatiotemporal dynamic behaviors of the model. Theoretical analysis shows that, in the absence of time delay, the nonlocal perception intensity η and perception range R can induce local over-predation of the host by natural enemies and population collapse, forming stable spatially nonhomogeneous steady-state solutions. When the parasitic time delay exceeds the critical value, the delay effect leads to periodic population oscillation. The synergistic effect of nonlocal perception and time delay induces Turing-Hopf bifurcation, presenting complex spatiotemporally coupled patterns. Numerical simulations based on parameters with realistic biological backgrounds verify the correctness of the theoretical analysis, and intuitively demonstrate the dynamic modes corresponding to different parameters, including local asymptotic stability, stable spatially nonhomogeneous steady-state solutions, and spatially nonhomogeneous periodic solutions. The results reveal the regulation mechanism of top-hat nonlocal perception and parasitic time delay on the spatiotemporal dynamics of the Monochamus saltuarius-Sclerodermus guani system. They not only enrich the spatiotemporal dynamics theory of facultative predation-parasitism systems, but also provide a quantitative mathematical basis for optimizing the release timing and release strategy of Sclerodermus guani in prevention and control as well as lay a theoretical foundation for the biological control of PWD.

In this paper, we propose a regional control strategy to address the challenge of cholera transmission, which is characterized by significant spatial heterogeneity due to its waterborne nature. A spatiotemporal dynamic model was formulated using reaction-diffusion equations, coupled with a four- dimensional optimal control framework encompassing awareness education, expanded vaccination, water chlorination, and treatment. The core mathematical analysis proceeded in three stages: First, the well-posedness of the state system was established, proving the existence and uniqueness of strong solutions. Second, the existence of an optimal control solution was rigorously demonstrated using the method of minimizing sequences. Finally, the first-order necessary optimality conditions were derived by constructing an adjoint system and applying Pontryagin’s maximum principle, which explicitly characterized the optimal control quadruple. Numerical simulations validated the theoretical framework, with direct comparisons between regional and uniform control strategies demonstrating the distinct advantage of our approach in cost-effectiveness and outbreak containment efficacy.

We propose a uniform weak Galerkin finite element method (WG-FEM) to solve singularly perturbed fourth-order convection–diffusion problems exhibiting boundary layers. The method is designed to handle small perturbation parameters ε, ensuring accurate resolution of sharp layers without spurious oscillations. We construct appropriate layer-adapted meshes and use uniform estimates to analyze the stability and convergence of the weak Galerkin scheme. Numerical experiments confirm the theoretical results and demonstrate that the method achieves uniform convergence with respect to the perturbation parameter, effectively capturing the boundary layer behavior on layer-adapted meshes. The proposed approach provides a reliable and efficient tool for high-order singularly perturbed problems with mixed derivative boundary conditions.

Compared with IP multicast, Overlay Multicast (OM) trees constructed at the application layer offer superior compatibility and flexible deployment advantages in heterogeneous, cross-domain networks. However, OM implementations under traditional network architectures suffer from weak adaptability to highly dynamic traffic due to their lack of awareness of underlying physical resource states. Moreover, reinforcement learning-based approaches fail to decouple the multi-objective tightly coupled nature of OM, resulting in high computational complexity, slow policy convergence, and insufficient stability. To address these challenges, we proposed a MA-DHRL-OM routing method. First, leveraging the centralized topological view provided by Software-Defined Networking (SDN), the method collected link-state information and constructed a traffic-aware feature model to provide multi-dimensional decision support for OM path planning. Second, within a unified framework that integrates multi-agent reinforcement learning and hierarchical reinforcement learning, MA-DHRL-OM solves for the optimal OM tree as follows: The hierarchical learning architecture decomposes the construction of the OM tree into a two-stage subtask framework. By designing tailored decision logic and reward signal feedback mechanisms for upper- and lower-layer agents, it achieved hierarchical decoupling of the high-dimensional OM problem, effectively reducing the action space dimensionality and enhancing policy convergence stability. Moreover, the multi-agent collaboration mechanism enabled each agent to make independent decisions based on its local observations, thereby balancing multi-objective optimization while improving the algorithm’s overall scalability and adaptability. Extensive simulation experiments demonstrated that, compared with existing methods, MA-DHRL-OM achieves superior performance in optimizing key metrics such as delay, bandwidth utilization, and packet loss rate while exhibiting more stable convergence behavior and greater flexibility in OM routing decisions.

This study proposes a predator-prey model that incorporates fear effects and additional food provision. A theoretical analysis is conducted by first establishing the positivity and boundedness of solutions, and then examining the existence and stability of all feasible equilibria. Models with unilateral and bilateral impulsive state feedback control are further introduced and analyzed. For these controlled systems, sufficient conditions for the existence and orbital asymptotic stability of order-1 periodic solutions are derived. Numerical simulations are provided to support the theoretical findings, showing excellent agreement with the analytical results. Finally, a systematic sensitivity analysis is performed to quantify the influence of the prey’s fear factor and the total amount of additional food on the system dynamics.

This paper introduced, for the first time, a novel discrete Langevin equation as a new gen- eralization of the classical Langevin model, involving two distinct integer-order differences. A new special function generated by the system parameters was defined to obtain explicit analytical solutions of the associated initial value problem. Several special cases were discussed, and open problems were proposed. The results established a new theoretical framework for discrete Langevin-type systems and contributed to the qualitative theory of difference equations.

This paper is devoted to the qualitative analysis of first-order nonlinear delay dynamic equations on time scales. We establish sufficient conditions for oscillation and nonoscillation of solutions to dynamic equations of the form

where α is a quotient of two odd integers andp, τ satisfy standard regularity and delay conditions on an arbitrary time scale T. Employing comparison principles, the time scales version of L’Hôpital’s rule, and integral transformations, we derive new oscillation criteria that generalize several classical results from the differential and difference equation settings to the unified framework of time scales. Furthermore, we provide sufficient conditions ensuring the existence of eventually positive (nonoscillatory) solutions. Our results extend and complement the existing theory in the literature.

In this paper, anS IRS epidemic model with periodic transmission and vaccination is proposed to study the periodic transmission phenomenon of influenza. The threshold dynamics of the model are completely determined by the basic reproduction number $\mathcal{R}_{0}$, i.e., the disease-free periodic steady state is globally asymptotically stable if $\mathcal{R}_{0}$ < 1, while the disease is uniformly persistent if $\mathcal{R}_{0}$ > 1. With the help of control theory, the resonance frequency ω_r of our model is obtained, which is numerically validated in that there exists a resonance phenomenon of the disease spread (i.e., there is a maximum amplitude of the oscillatory spread of diseases) when the frequency ω = ω_r. By fitting the real data of influenza from Shanghai, we estimate the parameter values of the model. Results show that there is no resonance phenomenon in the spread of influenza in Shanghai. Based on the expression of resonance frequency ω_r, the effects of the mean transmission rate β and the transition rate δ on the resonance frequency ω_r are displayed numerically.

In this paper, we defined affine Yano–Ricci solitons and perturbed Yano–Ricci solitons on three-dimensional Lorentzian Lie groups and obtain their complete classification. The major results indicated that all groups G₃, · · · ,G₇ admit both affine Yano–Ricci solitons and their perturbed companions, whereas G₁ supports neither. As a corollary, we found that the Lorentzian Lie groups G₃,G₅, and G₆ are affine Einstein manifolds with respect to the Yano connection.

In this paper, we study a structural acoustics model with supercritical source and nonlinear damping. The model consists of a wave equation defined on a bounded domain which is strongly coupled with a Berger plate equation acting on the elastic wall. The aim of the paper is to remove the strong condition $u \in L^{\infty}\left(\mathbb{R}^{+} ; L^{\frac{3}{2}}(\Omega)\right) \text { for } m_{1}>5$ in the work [1] to obtain a weaker energy decay, and give a logarithmic decay rate under a weak assumption on the initial data.

This paper investigates the anti-periodic boundary value problem for multi-term fractional differential equations that involve the p-Laplacian operator. We establish existence results by separately applying the Leray-Schauder nonlinear alternative and the Krasnosel’skii’s fixed point theorem. Finally, two illustrative examples are presented to demonstrate the effectiveness of the main results.